It happens like this: when you come to the end of the first page of Damon Galgut's superlative novel, In a Strange Room, you read its final sentence twice.

Applying his brush ever so deftly – a stroke of intent in a painterly work whose canvas stretches over three continents – the author writes: "What the first man is wearing I don't know, I forget."

The conundrum posed by this seemingly simple construction led Guardian literary editor Claire Armitstead to tweet, after In a Strange Room was longlisted for this year's Man Booker prize, wondering if the book should be classified as a novel at all. The reason: the man in the sentence and the person narrating it are the same. (Hence the reader's double-take.) This person is called "Damon", and we encounter him, like a god of ancient Egypt, in different forms. He is the Damon in the distance, the foreshortened "he" whom we crane our necks for a better glimpse of. He is the Damon in the room, the "I" who is reading the book alongside us, offering explanations and excuses. And he is the Damon in the mirror, the "you" to whom questions, accusations and truths about himself are put. Call it narrative syncretism.

Galgut the author deploys all three persons at any given place inside his book – including inside the same paragraph – and it's a tribute to his craft that, after the first jolt, the effect is remarkably turbulence-free. We cruise at 30,000 feet wholly enclosed by the stories that our companion, the 3-in-1 Damon, is telling us.

We're cruising, largely, outside South African airspace – unusual territory for Galgut, who is something of a South African landscape artist. His previous novel, The Impostor, is set in the Karoo, and the one before, The Good Doctor – also a Booker shortlistee – unfolds in a place that, while not named outright, strongly evokes the writer's homeland. Here, instead, we're in Greece, Lesotho, Malawi, Kenya, Switzerland and – in the novel's harrowing, unforgettable final section – India.

It's evident that Galgut is trying to escape from, or at least disengage with, South African identity politics in this book, and he has acknowledged as much in conversations following its release. His aim, he says, was not to present an identifiably South African point of view on the people and places he wrote about; rather, it was to convey what South African critic Chris Roper called, in his review of the novel, the "experiential economy" of the character, Damon, in the moment. Galgut does his utmost to background the passport on which Damon travels, seeking to tease out truths on a plane of higher, less constructed contingencies. It would seem that he has succeeded to a large extent in this, for the reviews of the book, inside South Africa at least, have avoided the phrase "South African novel" – although Roper extends a feeler in this direction, saying that Damon inhabits "a state of being that's extraordinarily relevant" to the country today. Sometimes, indeed, it feels like we're living inside a strange room here.

Galgut's not after any kind of universal truth, however: at no point is his book maudlin or grandiose. Rather, as he told me at the recent Cape Town Book Fair, he set himself the task of capturing mundane truths – the truths of what happened to Damon in his incarnations as The Follower, The Lover and The Guardian (as the book's three sections are called). Every sentence would be polished until it gleamed with a writer's truth – which, as any writer knows, is the gleam of the will-o-the-wisp. Calling the narrative fiction actually deepens its truth.

From South Africa, then, Galgut has a candid, vexing, but ultimately acceptable answer to Claire Armitstead's tweet: the events in In a Strange Room happened, but the book is not predicated on facts, it's predicated on truth, which is a function of art and art's inventions. Damon is an unforgettable invention, a character who couldn't be presented outside the confines of fiction, and the book that he inhabits, in my opinion, is as powerful and moving as any of the great works of fictional memoir – JM Coetzee's Boyhood and Paul Theroux's My Other Life spring immediately to mind. In a Strange Room presents us with a novelist at the height of his considerable powers and, in South Africa, we're pulling for him.